Motor having rotor of optimized shape

ABSTRACT

A motor according to an embodiment of the present invention includes a rotor having a rotor core and a plurality of magnetic poles including permanent magnets provided in the rotor core; and a stator having a stator core in which a plurality of teeth disposed on the side of the outer periphery of the rotor so as to be opposed to the plurality of magnetic poles and slots for containing armature winding wound around the plurality of teeth are formed. The rotor is structured such that the distance r(θ) between the center of the rotor and the outer periphery thereof satisfies the following equations (1) and (2): 
     
       
         
           
             
               
                 
                   
                     
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BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a motor, and specifically relates to amotor having a rotor of optimized shape.

2. Description of Related Art

It is generally known that the fundamental frequency component(hereinafter referred to as “fundamental component”) of a cogging torqueper rotation of a rotor, which can be reduced by optimizing the shape ofthe rotor, appears at a frequency of the least common multiple of thenumber of poles of the rotor and the number of teeth of a stator.

To reduce the fundamental component of a cogging torque, rotors havebeen conventionally optimized in shape in various ways. FIG. 1 shows avariation in the waveform (fundamental component) of a cogging torque,when a rotor is continuously varied in shape. Assuming that, when therotor is gradually varied in shape from (1) to (5), the cogging torquehas the lowest amplitude in the shape of (3). In this case, the shape of(3) is used as an optimal shape.

Conventional shape optimization aims at reducing a fundamentalcomponent, but cannot completely eliminate the fundamental component. Toeliminate the remaining fundamental component, in general, the shape ofa rotor, the shape of teeth of a stator, the phase relationship betweenthe rotor and the teeth, or the like is varied in the direction of amotor axis. As the most general method of them all, a skew structure inwhich the polar phase of a rotor is varied in an axial direction isknown (for example, Japanese Unexamined Patent Publication (Kokai) No.2014-150626, hereinafter referred to as “patent document 1”).

FIG. 2 is a schematic perspective view of a rotor having a two-layerskew structure in a conventional motor according to the patentdocument 1. A rotor 1002 has rotor core blocks 1041 a and 1041 b in twolayers in an axial direction. In each of the rotor core blocks 1041 aand 1041 b, permanent magnets 1050 are embedded to form a plurality ofmagnetic poles. The rotor 1002 has a layered skew structure in which therotor core blocks 1041 a and 1041 b are integrated into one unit in astate of being skewed relative to each other in a circumferentialdirection.

FIG. 3 is an enlarged view of an essential portion of a rotor core 1040of FIG. 2 viewed from the axial direction in the conventional motoraccording to the patent document 1. A skew angle is established suchthat in a rotor core block 1041 including the rotor core blocks 1041 aand 1041 a of two layers, at least part of flux barriers 1060 a and 1060b between adjacent magnetic poles overlap in space on both sides of themagnets 1050 fitted into magnet fitting recesses 1042. At least part ofthe overlapping flux barriers 1060 a and 1060 b are aligned across thelayers in the axial direction. The overlapping flux barriers 1060 a and1060 b have an approximately oval shape in cross section andapproximately the same outer peripheral shape. Approximately aligningthe flux barriers 1060 a and 1060 b across the layers intercepts a shortcircuit flux flowing through the layers in the axial direction.

In the conventional art of FIG. 2, the rotor is formed into a pluralityof layers, and the phase of the rotor is skewed relative to each layerto skew the phase relationship between the rotor and a stator. Thereby,a cogging torque occurs in different phases and cancels out owing tosuperposition. Besides the above, a method of skewing a stator phase, amethod of varying the shape of a rotor among layers, a method of varyingthe shape of teeth of a stator among layers, or the like may be used.

According to the conventional art, since the fundamental component of acogging torque is not sufficiently reduced, rotors sometimes have a skewstructure (continuous skew structure or skew structure among a pluralityof layers). However, the skew structure may cause a reduction in theoutput of motors. Also, the skew structure increases the number ofcomponents and man-hours, thus increasing costs.

SUMMARY OF THE INVENTION

The present invention aims at providing a motor in which, by optimizingthe shape of a rotor, the fundamental component of a cogging torqueassociated with the number of poles of the rotor and the number of slotsof a stator is eliminated to allow a significant reduction in thecogging torque.

A motor according to an embodiment of the present invention includes arotor having a rotor core and a plurality of magnetic poles includingpermanent magnets provided in the rotor core; and a stator having astator core in which a plurality of teeth disposed on the side of theouter periphery of the rotor so as to be opposed to the plurality ofmagnetic poles and slots for containing armature winding wound aroundthe plurality of teeth are formed. The rotor is structured such that thedistance r(θ) between the center of the rotor and the outer peripherythereof satisfies the following equations (1) and (2):

$\begin{matrix}{{\int_{- \phi}^{\phi}{{{{f(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \leq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{10}} & (1) \\{{\int_{- \phi}^{\phi}{{{{r_{1}(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} > 0} & (2)\end{matrix}$

wherein,

$\begin{matrix}{{f(\theta)} = {r_{0} - \frac{R - r_{0}}{\cos^{\alpha}\left( {\frac{\beta}{\mu^{r - 1}}{\theta }^{\gamma}} \right)}}} & (3) \\{{r_{1}(\theta)} = {r_{0} - \frac{R - r_{0}}{\cos\left( {\frac{p}{2}\theta} \right)}}} & (4) \\{\frac{3\pi}{5p} \leq \phi \leq \frac{\pi}{p}} & (5) \\{\frac{1}{3} \leq \alpha \leq 2} & (6) \\{\frac{p}{4} \leq \beta \leq p} & (7) \\{\frac{3\pi}{5p} \leq \mu \leq \frac{\pi}{p}} & (8) \\{\frac{1}{2} \leq \gamma \leq 4} & (9)\end{matrix}$

R: the minimum diameter of the stator core

r₀: the maximum diameter of the rotor

r₁: the diameter of a rotor of a conventional shape

p: the number of the poles of the rotor

θ: an angle [rad] with respect to a straight line (0 [rad]) that extendsfrom the rotation center of the rotor to the center of the pole of therotor orthogonal to a rotation axis.

ϕ: a specified range [rad] of r(θ)

α, β, γ, and μ: parameters each having a range specified by the aboveequations and characterizing the shape of the rotor.

BRIEF DESCRIPTION OF THE DRAWINGS

The objects, features, and advantages of the present invention will bemore apparent from the following description of an embodiment inconjunction with the attached drawings, wherein:

FIG. 1 is a schematic diagram showing a variation in the waveform(fundamental component) of a cogging torque, when a rotor iscontinuously varied in shape in a conventional motor;

FIG. 2 is a schematic perspective view of a rotor having a two-layerskew structure in the conventional motor;

FIG. 3 is an enlarged view of an essential portion of the conventionalmotor shown in FIG. 2 viewed from an axial direction;

FIG. 4 is a sectional view of a motor according to an embodiment of thepresent invention;

FIG. 5 is an enlarged sectional view of the periphery of a rotor in themotor according to the embodiment of the present invention;

FIG. 6 is a schematic diagram showing a variation in the waveform(fundamental component) of a cogging torque, when the rotor iscontinuously varied in shape in the embodiment of the present invention;

FIG. 7 is an enlarged sectional view of a magnet and a magnet recess inthe motor according to the embodiment of the present invention;

FIG. 8 shows simulation results of the shape of the rotor, whenparameters are changed, in the motor according to the embodiment of thepresent invention;

FIG. 9 is an enlarged sectional view of the magnet recess showing thedistance between the center of the rotor and the edge of the magnetrecess in the motor according to the embodiment of the presentinvention;

FIG. 10 shows a simulation result of a variation in the waveform of thefundamental component of a cogging torque, when the rotor iscontinuously varied in shape in the motor according to the embodiment ofthe present invention;

FIG. 11A is a graph showing the dependence of a cogging torque on arotation angle in the motor according to the embodiment of the presentinvention;

FIG. 11B is a graph showing the dependence of a cogging torque on arotation angle in the conventional motor;

FIG. 12A is a graph showing the frequency components of the coggingtorque in the motor according to the embodiment of the presentinvention;

FIG. 12B is a graph showing the frequency components of the coggingtorque in the conventional motor;

FIG. 13A is a graph showing a variation in the waveform of thefundamental component of the cogging torque, before shifting thepositions of magnets by 50 μm, in the motor according to the embodimentof the present invention;

FIG. 13B is a graph showing a variation in the waveform of thefundamental component of the cogging torque, after shifting thepositions of the magnets by 50 μm, in the motor according to theembodiment of the present invention; and

FIG. 14 is a graph showing variations in the waveforms of thefundamental component of the cogging torque, when the magnetic fluxdensity Br of the magnets is changed in the motor according to theembodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

A motor according to the present invention will be described below withreference to the drawings. FIG. 4 is a sectional view of the motoraccording to an embodiment of the present invention. A motor 100according to the embodiment of the present invention has a rotor 1 and astator 2. The rotor 1 has a rotor core 11 and a plurality of magneticpoles 13 including permanent magnets 12 provided in the rotor core 11.The stator 2 has a stator core 23 in which a plurality of teeth 21 andslots 22 are formed. The plurality of teeth 21 are disposed on the sideof the outer periphery of the rotor 1 so as to be opposed to theplurality of magnetic poles 13. The slots 22 contain armature winding(not shown) wound around the plurality of teeth 21.

In the motor 100 according to the embodiment of the present invention,the rotor 1 is structured such that the distance r(θ) between the centerC of the rotor 1 and the outer periphery of the rotor 1 satisfies thefollowing equations (1) and (2):

$\begin{matrix}{{\int_{- \phi}^{\phi}{{{{f(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \leq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{10}} & (1) \\{{\int_{- \phi}^{\phi}{{{{r_{1}(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} > 0} & (2)\end{matrix}$

where f(θ) is represented by the following equation (3):

$\begin{matrix}{{f(\theta)} = {r_{0} - \frac{R - r_{0}}{\cos^{\alpha}\left( {\frac{\beta}{\mu^{\gamma - 1}}{\theta }^{\gamma}} \right)}}} & (3) \\{{r_{1}(\theta)} = {r_{0} - \frac{R - r_{0}}{\cos\left( {\frac{p}{2}\theta} \right)}}} & (4)\end{matrix}$

In the equations (1) to (3), ϕ, α, β, μ, and γ are parameters thatcharacterize the shape of the rotor. The parameters are set so as tosatisfy the following equations (5) to (9):

$\begin{matrix}{\frac{3\pi}{5\; p} \leq \phi \leq \frac{\pi}{p}} & (5) \\{\frac{1}{3} \leq \alpha \leq 2} & (6) \\{\frac{p}{4} \leq \beta \leq p} & (7) \\{\frac{3\pi}{5\; p} \leq \mu \leq \frac{\pi}{p}} & (8) \\{\frac{1}{2} \leq \gamma \leq 4} & (9)\end{matrix}$

In the equations (1) to (3), R, r₀, r₁, p, θ, and ϕ are defined asfollows:

R: the minimum diameter of the stator core 23

r₀: the maximum diameter of the rotor 1

r₁: the diameter of a rotor of a conventional shape

p: the number of the poles of the rotor 1

θ: an angle [rad] with respect to a straight line (0 [rad]) that extendsfrom the rotation center C of the rotor 1 to the center of the pole ofthe rotor 1 orthogonally to a rotation axis.

ϕ: a specified range [rad] of r(θ)

FIG. 5 shows the relationship between f(θ) and r(θ). An area enclosedwith curves f(θ) and r(θ) is represented by the following equation overa range between −ϕ and ϕ. The left side of the equation (1) isproportional to the area.

$\frac{1}{2}{\int_{- \phi}^{\phi}{{{{f(\theta)}^{2} - {r(\theta)}^{2}}}\ d\;\theta}}$

The higher the left side of the equation (1), the larger the differencein shape between f(θ) and r(θ). On the other hand, the right side of theequation (1) is proportional to an area enclosed with the maximumdiameter r₀ of the rotor and the minimum diameter R of the stator, thatis, an area represented by the following equation:

$\frac{2{\phi\left( {R^{2} - r_{0}^{2}} \right)}}{2} = {\phi\left( {R^{2} - r_{0}^{2}} \right)}$

It is said that the equation (1) specifies “an allowance for thedifference between the curves f(θ) and r(θ) in the shape of the rotor”by a ratio relative to “the area enclosed with the maximum diameter r₀of the rotor and the minimum diameter R of the stator”.

The adequacy of a coefficient ( 1/10) on the right side of the equation(1) will be additionally described. By way of example, p=8, ϕ=π/8, R=30mm, r₀=29.6 mm, and the difference between the curves f(θ) and r(θ) isinvariably 20 μm. Assuming that f(θ) is circular for convenience incalculation, in FIG. 5, the area enclosed with the curves f(θ) and r(θ)over the range between −ϕ and ϕ is calculated as follows:

$\frac{1}{2}{\int_{- \phi}^{\phi}{{{{f(\theta)}^{2} - {r(\theta)}^{2}}}\ d\;\theta}}$=½·π/8·2·(29.6^2−29.58^2)≈0.4648

The area enclosed with the maximum diameter r₀ of the rotor and theminimum diameter R of the stator over the range between −ϕ and ϕ iscalculated as follows:

$\frac{2{\phi\left( {R^{2} - r_{0}^{2}} \right)}}{2} = {\phi\left( {R^{2} - r_{0}^{2}} \right)}$=π/8·(30^2−29.6^2)≈9.3619

The ratio between the above two calculation results is0.4648/90.3619=0.049648 . . . ≈ 1/20.

Therefore, the coefficient on the right side of the equation (1) isdetermined at 1/10.

$\begin{matrix}{{{\frac{1}{2}{\int_{- \phi}^{\phi}{{{{f(\theta)}^{2} - {r(\theta)}^{2}}}\ d\;\theta}}} \leq {\frac{1}{20}{\phi\left( {R^{2} - r_{0}^{2}} \right)}}}{{\int_{- \phi}^{\phi}{{{{f(\theta)}^{2} - {r(\theta)}^{2}}}\ d\;\theta}} \leq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{10}}} & (1)\end{matrix}$

Although f(θ) is not circular in actual fact, the assumption that f(θ)is circular is determined to be adequate to obtain an approximate valueof the area enclosed with the curves f(θ) and r(θ).

When similar calculation is performed with the assumption that thedifference between the curves f(θ) and r(θ) is invariably 10 μm, thefollowing equation is determined to be adequate.

${\int_{- \phi}^{\phi}{{{{f(\theta)}^{2} - {r(\theta)}^{2}}}\ d\;\theta}} \leq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{20}$

The parameter ϕ may sometimes be set so as to satisfy the followingequation (5′), instead of the equation (5).

$\begin{matrix}{\frac{4\pi}{5\; p} \leq \phi \leq \frac{\pi}{p}} & \left( 5^{\prime} \right)\end{matrix}$

The parameter α may sometimes be set so as to satisfy the followingequation (6′), instead of the equation (6).

$\begin{matrix}{\frac{1}{2} \leq \alpha \leq \frac{3}{2}} & \left( 6^{\prime} \right)\end{matrix}$

The parameter γ may sometimes be set so as to satisfy the followingequation (9′), instead of the equation (9).

$\begin{matrix}{\frac{1}{2} \leq \gamma \leq 2} & \left( 9^{\prime} \right)\end{matrix}$

The parameter β may sometimes be set so as to satisfy the followingequation (7′), instead of the equation (7).

$\begin{matrix}{\frac{p}{4} \leq \beta \leq \frac{p}{2}} & \left( 7^{\prime} \right)\end{matrix}$

The parameter ϕ may sometimes be set so as to satisfy the followingequation (5″), instead of the equation (5).

$\begin{matrix}{\frac{9\pi}{10\; p} \leq \phi \leq \frac{\pi}{p}} & \left( 5^{''} \right)\end{matrix}$

The parameter α may sometimes be set so as to satisfy the followingequation (6″), instead of the equation (6).α=1  (6″)

The parameter γ may sometimes be set so as to satisfy the followingequation (9″), instead of the equation (9).γ=1  (9″)

Depending on the accuracy of a mold, the distance r(θ) may be determinedso as to satisfy the following equation (1′), instead of the equation(1).

$\begin{matrix}{{\int_{- \phi}^{\phi}{{{{f(\theta)}^{2} - {r(\theta)}^{2}}}\ d\;\theta}} \leq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{20}} & \left( 1^{\prime} \right)\end{matrix}$

The distance r(θ) may be determined so as to satisfy the followingequation (2′), instead of the equation (2). In this case, the amount ofseparation from r₁ is made clearer.

$\begin{matrix}{{\int_{- \phi}^{\phi}{{{{r_{1}(\theta)}^{2} - {r(\theta)}^{2}}}\ d\;\theta}} \geq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{50}} & \left( 2^{\prime} \right)\end{matrix}$

The distance r(θ) may be determined so as to satisfy the followingequation (2″), instead of the equation (2). In this case, the amount ofseparation from r₁ is further made clearer.

$\begin{matrix}{{\int_{- \phi}^{\phi}{{{{r_{1}(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \geq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{20}} & \left( 2^{''} \right)\end{matrix}$

The distance r(θ) may be determined so as to satisfy the followingequations (10) and (20), instead of the equations (1) and (2).

$\begin{matrix}{{\int_{- \phi}^{\phi}{{{{f(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \leq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{20}} & (10) \\{{\int_{- \phi}^{\phi}{{{{r_{1}(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \geq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{20}} & (20)\end{matrix}$

Here, the equations (2) and (4) will be briefly described. A synchronousmotor is generally controlled on the precondition that a variation influx linkage with the rotation angle of a rotor is represented bytrigonometric functions. When the variation in flux linkage with therotation angle of the rotor is out of the trigonometric functions, atorque ripple (torque ripple during energization) is increased, thuscausing a deterioration of controllability. Thus, in order to make thevariation in flux linkage in the form of trigonometric functions,trigonometric functions are conventionally used for designing the shapeof the rotor itself (more specifically, the amount of variation in thedistance between the rotor and the stator is provided with trigonometricfunctions). At this time, one pole of the rotor corresponds to a halfperiod (one crest) of a trigonometric function (in the case of a p-polemotor, cos(pθ/2) is used).

In this embodiment, in order to optimize not a torque ripple but acogging torque, the shape of the rotor is optimized by a new method ofusing trigonometric functions, instead of a conventional method of usingtrigonometric functions (to put it the other way, the fundamentalcomponent of the cogging torque cannot be minimized by the conventionalmethod of using trigonometric functions). Therefore, the equations (2)and (4) are used, in contrast to the conventional method.

The equation (3) will be briefly described. In the equation (3), theparameters α, β, μ, and γ are added to the conventional functionalequation (4), in order to increase flexibility in the outer peripheralshape of the rotor and facilitate finding out the minimum point of thecogging torque. The above parameters and the outer peripheral shape ofthe rotor have the following relations.

-   -   When α is less than 1, f(θ) expands outside relative to the        outer peripheral shape of a rotor defined by the equation (4).        When α is more than 1, f(θ) contracts inside relative to the        outer peripheral shape of the rotor defined by the equation (4).    -   When β is less than p/2, f(θ) expands outside relative to the        outer peripheral shape of the rotor defined by the equation (4).        When β is more than p/2, f(θ) contracts inside relative to the        outer peripheral shape of the rotor defined by the equation (4).    -   When γ is less than 1, in a range of θ<μ, f(θ) contracts inside        relative to the outer peripheral shape of the rotor defined by        the equation (4). In a range of θ>μ, f(θ) expands outside        relative to the outer peripheral shape of the rotor defined by        the equation (4).

When γ is more than 1, in a range of θ<μ, f(θ) expands outside relativeto the outer peripheral shape of the rotor defined by the equation (4).In a range of θ>μ, f(θ) contracts inside relative to the outerperipheral shape of the rotor defined by the equation (4).

It is also to be noted that the manner of expanding the shape differsdepending on the parameters. “Expanded” or “contracted” areas are mainlyreferred to by a reference numeral 15 (shoulder portions of the poles)in FIG. 4, and the maximum diameter of the rotor does not change.

The minimum point of the cogging torque is found out using the abovecharacteristics and parameters including the size and positions of themagnets.

In the motor according to the embodiment of the present invention, theouter peripheral shape of the rotor is defined as shown in FIG. 4 andoptimized by varying one or a plurality of parameters of the functionalparameters α, β, γ, and μ, the positions of the magnets, the shape ofthe magnets, and the shape of magnet recesses (in the case of an IPM(interior permanent magnet) motor). This brings about different effectsfrom those obtained by conventional optimization. The shape of the rotorin the motor according to the embodiment of the present inventionincludes all of the parameters described above, not only the outerperipheral shape but also all of the positions of the magnets, the shapeof the magnets, and the shape of the magnet recesses.

Moreover, the shape of the rotor 1, the shape of the teeth 21 of thestator 2, and the phase relationship between the rotor 1 and the teeth21 are preferably invariant in an axial direction, and the plurality ofmagnetic poles 13 of the rotor 1 preferably have the same shape. Thisstructure eliminates the need for providing a skew or a step stagger,thus preventing a reduction in torque owing thereto.

The motor according to the embodiment of the present invention focuseson the fact that when the rotor is continuously varied in shape, thephase of the fundamental component is shifted by a half period(reversed). The fundamental component is cancelled using the shape atthe time of reversing. The outer peripheral shape of the rotor isrepresented by trigonometric functions. The outer peripheral shape ofthe rotor is varied and optimized using the plurality of parameters.

FIG. 6 is a diagram showing a variation in the waveform (fundamentalcomponent) of a cogging torque, when the rotor is continuously varied inshape. In the motor according to the embodiment of the presentinvention, the above-described plurality of parameters (or one of them)are gradually varied as shown at rotor shape numbers (1) to (7). Whenthe parameters are gradually varied, as shown in FIG. 6, the fundamentalcomponent of the cogging torque is varied and, when compared between (1)and (7), the phase of the fundamental component is shifted by a halfperiod (the waveform is reversed). Therefore, there is a shape having afundamental component of infinitesimally close to 0 in the middlebetween (1) and (7), for example, at (4). Thus, using the shape having acogging torque of close to 0 (for example, the shape of (4)) allowsproducing a motor that has an extremely small fundamental component.

The outer peripheral shape of the rotor refers to the outer peripheralshape of magnets in surface permanent magnet (SPM) motors, while refersto the outer peripheral shape of a rotor core in interior permanentmagnet (IPM) motors. The shape of a magnetic substance (the shape of asubstance that has an effect on a cogging torque) is primarily kept inmind, though the shapes of a non-magnetic substance and the like (theshape of a substance that has no effect on a cogging torque, such as aresin cover and a thin SUS cover) are not kept in mind.

The equation (4) is a function that is conventionally used for designingthe outer shape of a rotor, and is a general method of usingtrigonometric functions. The equation (2) clearly specifies a differentshape from the equation (4). For the sake of more accurate optimization,the outer peripheral shape is intentionally deviated from generaltrigonometric functions using the various parameters, as represented bythe equation (3).

To make the difference between the outer peripheral shape and theconventional shape outstand, a value is substituted into the right sideof the equation (2), as follows:

$\begin{matrix}{{\int_{- \phi}^{\phi}{{{{r_{1}(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \geq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{100}} & (9) \\{{\int_{- \phi}^{\phi}{{{{r_{1}(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \geq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{10}} & (10)\end{matrix}$

The shape represented by r(θ), which contributes to a torque, differsmore from a conventional shape r₁ in the equation (10) than in theequation (9).

FIG. 7 is an enlarged sectional view of a magnet and a magnet recess inthe motor according to the embodiment of the present invention. As shownin FIG. 7, the shape of a magnet recess 14 may not be a simple recessfor containing the magnet 12 but have projections that are differentfrom the shape of the magnet 12 itself, as indicated by A and A′ in FIG.7.

Next, examples of the outer peripheral shape of the rotor and advantagesthereof will be described. FIG. 8 shows simulation results of the shapeof the rotor with changes in the parameters in the motor according tothe embodiment of the present invention.

In FIG. 8, a curve (i) is a plot of the following equation (4), which isconventionally used for designing the shape of the rotor.

$\begin{matrix}{{r_{1}(\theta)} = {r_{0} - \frac{R - r_{0}}{\cos\left( {\frac{p}{2}\theta} \right)}}} & (4)\end{matrix}$

Curves (ii) to (iv) represent the outer peripheral shapes of the rotor,when the various parameters of the equation (3) are set as follows:

(ii) α=1, γ=1, β=3.8, μ=1

(iii) α=1, γ=1.7, β=7.5, μ=1

(iv) α=1, γ=1, β=3.5, μ=1

In FIG. 8, rectangles shown by dotted lines represent examples of thedisposition of the magnet recess.

It is found out that the outer peripheral shapes of the rotor in thecases of (ii) to (iv) have a higher degree of flexibility in the size ofthe magnet and the position of the magnet recess than the outerperipheral shape of the rotor in the case of (i). Thus, it is possibleto prevent a reduction in a rated output owing to the optimization ofthe shape of the rotor. The higher degree of flexibility in thepositions of the magnet recesses facilitates fine adjustment of thepositions and shape of the magnet recesses, thus serving to minimize thecogging torque.

FIG. 9 is an enlarged sectional view of the magnet recess showing thedistance between the center of the rotor and the edge of the magnetrecess in the motor according to the embodiment of the presentinvention. In FIG. 9, r_(m) is the distance between the center C of therotor and the midpoint of the outer edge of the magnet recess, r_(mm) isthe maximum value of the distance between the center C of the rotor andthe outer edge of the magnet recess in the vicinity of an interpolarportion, and r_(g) is the distance between the center C of the rotor andthe center of the interpolar portion.

In this embodiment, r_(mm)>r_(g) basically stands true. Thus, bynarrowing the distance between the magnet recess and the outer peripheryof the rotor core, magnetic saturation occurs in areas X and X′ in FIG.9, thus resulting in an increase in a flux linkage owing to the magnet.At the same time, the cogging torque is optimized using the variousparameters. As necessary, the cogging torque may be optimized usingr_(g) as another parameter. In this case, irrespective of r_(mm)>r_(g),r_(g) may be increased and, for example, a magnetic saturation portionmay be formed from an interpolar straight portion and the magnet recessto optimize the cogging torque.

Next, the continuity of a variation in the shape of the rotor and thecontinuity of the waveform of the cogging torque will be described. As amatter of course, even if the rotor and the teeth of the stator have anyshape (the outer peripheral shape, the positions of the magnets, and thelike), continuity is insured between different shapes. In other words,the rotor can be changed to a completely different shape by a gradualvariation in shape.

It is conceivable that, with the gradual variation in shape, thewaveform of the cogging torque is continuously varied, and it actuallyis. There is no singular point in the continuous variation in shape in asensible range. If the cogging torque is reversed between differentshapes of the rotor, it is assumed due to the continuity that there is ashape having no cogging torque between the shapes.

Note that, the “sensible range” described here refers to a range inwhich the magnet does not contact the magnet next thereto or the outerperiphery of the rotor does not contact the stator, in other words, arange in which a close surface constituting the rotor does not makecontact during a variation in shape.

FIG. 10 shows a simulation result of a cogging torque with a gradual andcontinuous variation in the shape of the rotor from (1) to (6) in an 8pole 36 slot motor (the number of poles is 8, and the number of slots is36). In each shape, a simulation is performed by a 1/72 turn (5degrees). That is to say, since the fundamental frequency component ofthe cogging torque produced by the 8 pole 36 slot motor appears at afrequency of 72, the simulation is performed for one period.

Comparing the simulation result at (1) and (6) in FIG. 10, a phase isshifted by a half period (a waveform is reversed). At (4) between (1)and (6), a wave of the fundamental frequency component is approximately0. A waveform of (1) to (6) is gradually and continuously varied.

Next, results of a comparison between a motor according to the presentinvention and a motor optimized according to the conventional art, whichare actually manufactured, will be described. FIGS. 11A and 11B showmeasurement results of a cogging torque in the motors optimizedaccording to the present invention and according to the conventionalart, respectively, for the purpose of comparison. FIGS. 12A and 12B showthe frequency components of the cogging torque in the motors optimizedaccording to the present invention and according to the conventionalart, respectively, for the purpose of comparison.

It is apparent from FIGS. 11A, 11B, 12A, and 12B that the motoroptimized according to the present invention has a much lowerfundamental component (this motor has a frequency of 72 per rotation) ofthe cogging torque than the motor optimized according to theconventional art.

In a cogging torque per rotation of a rotor, it should be consideredthat, even if the amplitude of the cogging torque of a frequencycomponent at a frequency of the least common multiple of the number ofpoles of the rotor and the number of teeth of a stator is optimized (byshape optimization), the cogging torque possibly remains in the order of0.125% of a rated torque, at the least. When a motor is actuallymanufactured, each of magnets, the stator, and the rotor are required tohave a dimensional tolerance in shape. Also, the Br (magnetic fluxdensity) of the magnets is required to have a tolerance. Thus, even ifthe fundamental component of a cogging torque is completely cancelled ina simulation, the actual motor has accuracy limitations and a smallfundamental component of the cogging torque remaining. Since a plot isnot a perfectly flat straight line even in the simulation, and acalculation error in the simulation has to be taken in account, it isappropriate to think that the actual motor has a slight fundamentalcomponent remaining, even if an ideal shape is intended to be chosen.

Thus, the fundamental component remains as follows:

The fundamental component contained in the shape itself (including asimulation error): 0.025%

Influence by the positions and shape tolerance of the magnets: 0.025%

Influence by the shape tolerance of the rotor core: 0.025%

Influence by the shape tolerance of the teeth of the stator: 0.025%

Influence by the tolerance of the Br of the magnets: 0.025%

The above fundamental components sum to 0.125%.

The above are values under relatively advanced technical management. Inthe case of processing or a choice of the magnets requiring a low cost,a quick delivery, or the like, a larger fundamental component (forexample, 0.2%) possibly remains, even after the shape optimization,depending on circumstances.

FIGS. 13A and 13B show variations (simulation results) in the waveformof the fundamental component of a cogging torque, before and aftershifting the positions of magnets of a rotor (of an IPM motor) by 50 μm,respectively. By shifting the magnets by 50 μm, an amplitude ((maximumvalue-minimum value)/2) is increased by 0.073% from 0.123% to 0.196%.This corresponds to 0.0146% per 10 μm by simple calculation. Consideringa processing technique and a yield rate, since a tolerance of the orderof ±10 to 20 μm is appropriately used at the least, so that it should beconsidered that the cogging torque varies in the order of 0.0146 to0.0292%. Thus, the above estimation value of 0.025% owing to thepositions and shapes of the magnets is considered to be appropriate (nottoo large).

The shape of the rotor and the shape of the teeth of the stator have ahigh degree of flexibility within the tolerance, and seem to have moreinfluence on the cogging torque than the positions of the magnets in theIPM motor. However, in the estimation that a variation of 0.025% isrequired at the least, a value of 0.025%, which is equal to the valueassociated with the positions of the magnets, is used.

FIG. 14 shows variations in the waveforms of the fundamental componentof a cogging torque, when the magnetic flux density Br of the magnets ischanged by ±2%. When the magnetic flux density Br of the magnets isshifted from a specified value by +2%, the amplitude ((maximumvalue-minimum value)/2) of the cogging torque deteriorates byapproximately 0.02%. Since industrially marketed magnets often have a Brtolerance of ±2 to ±3%, the above estimation value of 0.025% isappropriate.

In conventional shape optimization, the shape of a rotor is sometimesvaried or the phase of the rotor is sometimes varied (with a skew, alayered skew, or the like) in an axial direction of the rotor, in orderto cancel the fundamental component of a cogging torque. In such aninstance, the complicated structure causes an increase in costs, areduction in output, and the like.

According to the present invention, a rotor is formed in anunconventional shape that serves to reduce the fundamental component ofa cogging torque, thus allowing a reduction of the cogging torque.

When the shape optimization is insufficient, the fundamental componentis not completely canceled in most cases, even if the rotor is optimallyskewed (specifically not in the case of a continuous skew structure butin the case of a layered skew structure). Therefore, by the optimizationof the cogging torque, even if the fundamental component slightly occursowing to a processing tolerance, an assembly error, and the like, theprovision of a skew structure can substantially completely cancel thefundamental component of the cogging torque.

According to the motor of the embodiment of the present invention, byoptimizing the shape of the rotor, the fundamental component of thecogging torque associated with the number of the poles of the rotor andthe number of the slots of the stator is eliminated, thus allowing asignificant reduction in the cogging torque.

What is claimed is:
 1. A motor comprising: a rotor having a rotor coreand a plurality of magnetic poles including permanent magnets providedin the rotor core; and a stator having a stator core in which aplurality of teeth disposed on the side of the outer periphery of therotor so as to be opposed to the plurality of magnetic poles and slotsfor containing armature winding wound around the plurality of teeth areformed, wherein the rotor has an outer peripheral surface, the outerperipheral surface having a curved shape with shape parameters, α, β, γ,and μ as set forth below, such that a distance r(θ) between the centerof the rotor and the outer peripheral surface is related to a firstfunction ƒ(θ) describing a first arcuate shape, and a second functionr₁(θ) describing a second arcuate shape by the following equations (1)and (2): $\begin{matrix}{{\int_{- \phi}^{\phi}{{{{f(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \leq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{10}} & (1) \\{{\int_{- \phi}^{\phi}{{{{r_{1}(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} > 0} & (2)\end{matrix}$ wherein, $\begin{matrix}{{f(\theta)} = {r_{0} - \frac{R - r_{0}}{\cos^{\alpha}\left( {\frac{\beta}{\mu^{\gamma - 1}}{\theta }^{\gamma}} \right)}}} & (3) \\{{r_{1}(\theta)} = {r_{0} - \frac{R - r_{0}}{\cos\left( {\frac{p}{2}\theta} \right)}}} & (4) \\{\frac{3\;\pi}{5p} \leq \phi \leq \frac{\pi}{p}} & (5) \\{\frac{1}{3} \leq \alpha \leq 2} & (6) \\{\frac{p}{4} \leq \beta \leq p} & (7) \\{\frac{3\;\pi}{5p} \leq \mu \leq \frac{\pi}{p}} & (8) \\{\frac{1}{2} \leq \gamma \leq 4} & (9)\end{matrix}$ R: the minimum diameter of the stator core r₀: the maximumdiameter of the rotor p: the number of poles of the rotor θ: an angle[rad] with respect to a straight line (0 [rad]) that extends from therotation center of the rotor to the center of the pole of the rotororthogonally to a rotation axis ϕ: a specified range [rad] of r(θ) α, β,γ, and μ: parameters each having a range specified by the equations(6)-(9) wherein μ⁰=1.
 2. The motor according to claim 1, wherein in acogging torque per rotation of the rotor, the amplitude of the coggingtorque of a frequency component at a frequency of the least commonmultiple of a rotor pole number and a stator teeth number is 0.2% orless of a rated torque.
 3. The motor according to claim 1, wherein in acogging torque per rotation of the rotor, the amplitude of the coggingtorque of a frequency component at a frequency of the least commonmultiple of a rotor pole number and a stator teeth number is 0.125% orless of a rated torque.
 4. The motor according to claim 1, wherein theshape of the rotor, the shape of the teeth of the stator, and the phaserelationship between the rotor and the teeth are invariant in an axialdirection, and the plurality of magnetic poles of the rotor have thesame shape.
 5. The motor according to claim 1, wherein the parameter ϕis determined so as to satisfy the following equation (5′), instead ofthe equation (5) $\begin{matrix}{\frac{4\;\pi}{5p} \leq \phi \leq {\frac{\pi}{p}.}} & \left( 5^{\prime} \right)\end{matrix}$
 6. The motor according to claim 1, wherein the parameter αis determined so as to satisfy the following equation (6′), instead ofthe equation (6) $\begin{matrix}{\frac{1}{2} \leq \alpha \leq {\frac{3}{2}.}} & \left( 6^{\prime} \right)\end{matrix}$
 7. The motor according to claim 1, wherein the parameter γis determined so as to satisfy the following equation (9′), instead ofthe equation (9) $\begin{matrix}{\frac{1}{2} \leq \gamma \leq 2.} & \left( 9^{\prime} \right)\end{matrix}$
 8. The motor according to claim 1, wherein the parameter βis determined so as to satisfy the following equation (7′), instead ofthe equation (7) $\begin{matrix}{\frac{p}{4} \leq \beta \leq {\frac{p}{2}.}} & \left( 7^{\prime} \right)\end{matrix}$
 9. The motor according to claim 1, wherein the parameterϕ, is determined so as to satisfy the following equation (5″), insteadof the equation (5) $\begin{matrix}{\frac{9\;\pi}{10\; p} \leq \phi \leq {\frac{\pi}{p}.}} & \left( 5^{''} \right)\end{matrix}$
 10. The motor according to claim 1, wherein the parameterα is determined so as to satisfy the following equation (6″), instead ofthe equation (6)α=1  (6′).
 11. The motor according to claim 1, wherein the parameter γis determined so as to satisfy the following equation (9″), instead ofthe equation (9)γ=1  (9″).
 12. The motor according to claim 1, wherein the distance r(θ)is determined so as to satisfy the following equation (1′), instead ofthe equation (1) $\begin{matrix}{{\int_{- \phi}^{\phi}{{{{f(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \leq {\frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{20}.}} & \left( 1^{\prime} \right)\end{matrix}$
 13. The motor according to claim 1, wherein the distancer(θ) is determined so as to satisfy the following equation (2′), insteadof the equation (2) $\begin{matrix}{{\int_{- \phi}^{\phi}{{{{r_{1}(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \geq {\frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{50}.}} & \left( 2^{\prime} \right)\end{matrix}$
 14. The motor according to claim 1, wherein the distancer(θ) is determined so as to satisfy the following equation (2″), insteadof the equation (2) $\begin{matrix}{{\int_{- \phi}^{\phi}{{{{r_{1}(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \geq {\frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{20}.}} & \left( 2^{''} \right)\end{matrix}$
 15. The motor according to claim 1, wherein the distancer(θ) is determined so as to satisfy the following equations (10) and(20), instead of the equations (1) and (2) $\begin{matrix}{{\int_{- \phi}^{\phi}{{{{f(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \leq \frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{20}} & (10) \\{{\int_{- \phi}^{\phi}{{{{r_{1}(\theta)}^{2} - {r(\theta)}^{2}}}d\;\theta}} \geq {\frac{\phi\left( {R^{2} - r_{0}^{2}} \right)}{20}.}} & (20)\end{matrix}$